The plane wall in Figure $3$ is constructed from three materials with thermal conductivities $k_A,k_B$ and $k_B$ and thicknesses $L_A,L_B$ and $L_C,$ respectively. A contact resistance exists at the interface between material B and C$,$ no contact resistance exits between materials A and B$.$ The left side of Material A $($see figure$)$ is maintained at $T_{s,1}$ while material C adjoins a fluid at a temperature $T_{}$ with a convective heat transfer coefficient $h.$ The wall height is given by $H,$ the width by W and the area by A$.$

$($a$)$ Draw the equivalent resistance circuit, label the nodes, and equivalent resistances $(5$ points$)($b$)$ Find the algebraic expression denoting the heat transfer q through the wall in terms of given symbolic parameters $(10$ points$)($c$)$ Using the solution from part $($b$)$ Calculate the heat transfer rate through the wall if $k_A=0\mathrm{.}07\frac{W}{m}K,k_B=0\mathrm{.}05\frac{W}{m}K,k_C=0\mathrm{.}03\frac{W}{m}K,L_A=11mm,L_B=13mm,L_C=9mm{T}_{}=20C,T_{s,1}=175C,h=12\frac{W}{m^2}K,H=2\mathrm{.}0m$ and $W=2\mathrm{.}5m$ and, the contact resistance between wall B and C is $R_c=0\mathrm{.}03\frac{K}{W}(5$ points$).$